Magnetic properties of ${\text{LiCu}}_3{\text{O}}_3$: A quasi-two-dimensional antiferromagnet on a depleted square lattice

${\mathrm{LiCu}}_3{\mathrm{O}}_3$ is a novel two-dimensional $S=1/2$ antiferromagnet with randomly depleted square lattice. The crystal structure contains two types of square planes with different ${\mathrm{Cu}}^{2+}\to {\mathrm{Li}}^{+}$ substitution rates (20% and 40%). ${}^7\mathrm{Li}$ NMR and magnetization measurements performed on single crystals of ${\mathrm{LiCu}}_3{\mathrm{O}}_3$ revealed the occurrence of magnetic order at $T_{c1}=123\mathrm{K}$ and the change of the magnetic state at $T_{c2}\approx 30\mathrm{K}$. The high-temperature transition can be attributed to establishment of magnetic order in planes with higher concentration of magnetic ions and the low-temperature transition to the magnetic ordering in planes with lower concentration of magnetic ions. Broad continuous NMR spectra below $T_{c1}$ reflect a continuous distribution of values or directions of magnetic moments in ${\mathrm{LiCu}}_3{\mathrm{O}}_3$ typical for spiral, spin-modulated magnetic structures or structures with frozen disorder. Magnetization measurements revealed a spin-flop transition which indicates weak uniaxial anisotropy of the spin structure. The relatively small magnetic susceptibility at all orientations of applied magnetic field shows that the magnetic structure is rigid since the estimated value of saturation field derived from differential susceptibility measured at ${\mu }_{0}H=5\mathrm{T}$ is ${\mu }_0{H}_{\mathrm{sat}}\approx 200\mathrm{T}$.

Figure 1

Crystal structure of ${\mathrm{LiCu}}_3{\mathrm{O}}_3$. The smaller orange spheres mark the positions of the ${\mathrm{Cu}}^{+}$ nonmagnetic ions, while the smaller magenta spheres in B, A, B square planes mark the positions of randomly distributed ions of nonmagnetic ${\mathrm{Li}}^{+}$ and magnetic ${\mathrm{Cu}}^{2+}$. The larger blue spheres are at the positions of the ${\mathrm{O}}^{2-}$ ions.

Figure 2

Fragments of simulated square planes A and B with random occupation of crystal nodes of a regular square lattice by magnetic ions ${\mathrm{Cu}}^{2+}$ (crosses) and nonmagnetic ${\mathrm{Li}}^{+}$ (circles). The bonds between nearest magnetic ions are shown with lines. Ions which have all four nearest magnetic neighbors are highlighted with bold red crosses. The ion which has no in-plane magnetic neighbors is highlighted with a bold blue cross.

Figure 3

(Top panel) Temperature dependence of magnetic susceptibility $\chi \left(T\right)$ of ${\mathrm{LiCu}}_3{\mathrm{O}}_3$ measured at ${\mu }_{0}H=1\mathrm{T}$ (blue line) and ${\mu }_{0}H=5\mathrm{T}$ (red line), $\mathbf{H}\parallel C_4$. Red squares show temperature dependence of the value of differential susceptibility obtained from $M\left(H\right)$ measurements at ${\mu }_{0}H=5\mathrm{T}$ (see Fig. 5). Inverse susceptibilities $1/\chi \left(T\right)$ are shown in the inset to the Figure with respective colors. (Bottom panel) Temperature derivatives of $\chi \left(T\right)$ from the top panel. In the inset an expanded area in the vicinity of the singularity observed at $T\approx 120\mathrm{K}$ is presented. Green solid lines mark the smoothed slopes of $\chi \left(T\right)$ at both sides of the critical point.

Figure 4

(Top panel) Temperature dependence of magnetic susceptibility $\chi \left(T\right)$ of ${\mathrm{LiCu}}_3{\mathrm{O}}_3$ measured at ${\mu }_{0}H=1\mathrm{T}$ (blue line) and ${\mu }_{0}H=5\mathrm{T}$ (red line), $\mathbf{H}\perp C_4$. Red squares show temperature dependence of the value of differential susceptibility obtained from $M\left(H\right)$ measurements at ${\mu }_{0}H=5\mathrm{T}$ (see Fig. 6). Inverse susceptibilities $1/\chi \left(T\right)$ are shown in the inset to the figure with respective colors. (Bottom panel) Temperature derivatives of $\chi \left(T\right)$ from top panel. In the inset an expanded area in the vicinity of the singularity observed at $T\approx 120\mathrm{K}$ is presented. Green solid lines mark the smoothed slopes of $\chi \left(T\right)$ at both sides of the critical point.

Figure 5

(Top panel) Field dependence of magnetization $M$ of ${\mathrm{LiCu}}_3{\mathrm{O}}_3$ measured at different temperatures, $\mathbf{H}\parallel C_4$. (bottom panel) $M\left(H\right)$ measured at $T=1.85\mathrm{K}$ (black line) and at $T=30\mathrm{K}$ (green line). Linear and nonlinear parts of $M(H,T=1.85\mathrm{K})$ obtained as described in the text are shown with dashed red and blue lines, respectively.

Figure 6

(Top panel) Field dependence of magnetization $M$ of ${\mathrm{LiCu}}_3{\mathrm{O}}_3$ measured at different temperatures, $\mathbf{H}\parallel \left[1\overline{1}0\right](\perp C_4)$. (bottom panel) $M\left(H\right)$ measured at $T=1.85\mathrm{K}$ (black line) and at $T=30\mathrm{K}$ (green line). Linear and nonlinear parts of $M(H,T=1.85\mathrm{K})$ obtained as described in the text are shown with dashed red and blue lines, respectively.

Figure 7

(Left panel) Angle dependence of magnetization $M\left(\alpha \right)$ of ${\mathrm{LiCu}}_3{\mathrm{O}}_3$ measured at different values of ${\mu }_{0}H, T=1.85\mathrm{K}$. (Right panel) Angle dependence of magnetization $M\left(\alpha \right)$ of ${\mathrm{LiCu}}_3{\mathrm{O}}_3$ measured at different temperatures $T, {\mu }_{0}H=5\mathrm{T}$.

Figure 8

(Left panel) Angle dependencies of magnetic moment $M$ of ${\mathrm{LiCu}}_3{\mathrm{O}}_3$ measured at ${\mu }_{0}H=1-5\mathrm{T}, T=1.85\mathrm{K}$. (Middle panel) (a) $M\left(\phi \right)$ measured at ${\mu }_{0}H=2\mathrm{T}$. (b) $\pi$ harmonic of $M(\phi ,{\mu }_{0}H=2\mathrm{T})$ harmonic series. (c) Results of subtraction of $\pi$ harmonic from $M(\phi ,{\mu }_{0}H=2T)$. (d) Angle derivative of $M(\phi ,{\mu }_{0}H=2\mathrm{T})-M^{\pi }(\phi ,{\mu }_{0}H=2T)$. (Right panel) Results of subtraction of second harmonics $M^{\pi }\left(\phi \right)$ from magnetization measurements $M\left(\phi \right)$. Arrows at the curves in the figure show the directions of field rotations.

Figure 9

Temperature evolution of ${}^7\mathrm{Li}$ NMR spectra at $\mathbf{H}$ parallel and perpendicular to the crystallographic axis $C_4$; ${\mu }_0{H}_0=1.813\mathrm{T}$. NMR lines are stacked along ordinate axis for clarity.

Figure 10

Temperature dependence of ${}^7\mathrm{Li}$ NMR spectra linewidth $\delta \nu \left(T\right)$ measured at 1/2 (red symbols) and 1/3 (black symbols) of maximum. Squares correspond to $\mathbf{H}\parallel C_4$ and triangles correspond to $\mathbf{H}\perp C_4$. Transparent lines are given as guides for eye. Solid lines present the fit to the power-law behavior with exponents $\beta =0.5$ for $\mathbf{H}\parallel C_4$ and $\beta =0.25$ for $\mathbf{H}\perp C_4$ and $T_{c1}=123\mathrm{K}$.

Figure 11

(Left panel) Possible orientations of antiferromagnetic vector of model collinear structure at $H=0$. (Right panel) Schema of a model two-domain structure with dominant easy-plane and weak in-plane anisotropies at field applied in easy plane. Angles are counted from the equilibrium direction of $\mathbf{l}$ at $H=0$. Here, $\mathbf{a},\mathbf{b}$ mark the easy axes in plane (001).

Figure 12

The results of modeling ${}^7\mathrm{Li}$ NMR spectra, $\mathbf{H}\perp C_4$ from Fig. 9. Black lines show experimental spectra obtained at $T=4.2\mathrm{K}$ (top panel) and $T=47.4\mathrm{K}$ (bottom panel). Corresponding red lines show spectra calculated in frames of the models described in text. Insets show the model orientation of magnetic moments in crystallographic planes A (blue circles) and B (magenta circles).